Singular shocks, or solutions of very low regularity to systems of conservation laws, were first described in the 1980's by Herbert Kranzer and myself. They appeared in solving Riemann problems with large data in a strictly hyperbolic system of two conservation laws, and appeared to be a mere curiosity. More recently, they have appeared in some systems of physical interest, such as models for two-phase flow and a non-standard model for chromatography. The new examples, unlike the original one, all exhibit loss of hyperbolicity in regions of phase space, as well as the potential to allow these low-regularity waves.

This talk will explain the background of singular shocks, describe some ways of approximating them, and outline a theory that has been developed for hyperbolic models. I will also try to explain where non-hyperbolic models come from, why it may be useful to study them further, and how singular shocks may help in analysing their solutions.